A note on the Bloch-Tamagawa space and Selmer groups
aa r X i v : . [ m a t h . N T ] J a n A NOTE ON THE BLOCH-TAMAGAWA SPACE AND SELMER GROUPS
NIRANJAN RAMACHANDRANA
BSTRACT . For any abelian variety A over a number field, we construct an extension of the Tate-Shafarevich group by the Bloch-Tamagawa space using the recent work of Lichtenbaum and Flach.This gives a new example of a Zagier sequence for the Selmer group of A . Introduction.
Let A be an abelian variety over a number field F and A ∨ its dual. B. Birch andP. Swinnerton-Dyer, interested in defining the Tamagawa number τ ( A ) of A , were led to theircelebrated conjecture [2, Conjecture 0.2] for the L-function L ( A, s ) (of A over F ) which predictsboth its order r of vanishing and its leading term c A at s = 1 . The difficulty in defining τ ( A ) directly is that the adelic quotient A ( A F ) A ( F ) is Hausdorff only when r = 0 , i.e., A ( F ) is finite. S. Bloch[2] has introduced a semiabelian variety G over F with quotient A such that G ( F ) is discrete andcocompact in G ( A F ) [2, Theorem 1.10] and famously proved [2, Theorem 1.17] that the Tamagawanumber conjecture - recalled briefly below, see (5) - for G is equivalent to the Birch-Swinnerton-Dyer conjecture for A over F . Observe that G is not a linear algebraic group. The Bloch-Tamagawaspace X A = G ( A F ) G ( F ) of A/F is compact and Hausdorff.The aim of this short note is to indicate a functorial construction of a locally compact group Y A (1) → X A → Y A → Ш ( A/F ) → , an extension of the Tate-Shafarevich group Ш ( A/F ) by X A . The compactness of Y A is clearlyequivalent to the finiteness of Ш ( A/F ) . This construction would be straightforward if G ( L ) werediscrete in G ( A L ) for all finite extensions L of F . But this is not true (Lemma 4): the quotient G ( A L ) G ( L ) is not Hausdorff, in general. The very simple idea for the construction of Y A is: Yoneda’s lemma . Namely, we consider thecategory of topological G -modules as a subcategory of the classifying topos B G of G (natural fromthe context of the continuous cohomology of a topological group G , as in S. Lichtenbaum [10],M. Flach [5]) and construct Y A via the classifying topos of the Galois group of F .D. Zagier [18] has pointed out that the Selmer groups Sel m ( A/F ) (6) can be obtained fromcertain two-extensions (7) of Ш ( A/F ) by A ( F ) ; these we call Zagier sequences. We show how Y A provides a new natural Zagier sequence. In particular, this shows that Y A is not a split sequence.Bloch’s construction has been generalized to one-motives; it led to the Bloch-Kato conjectureon Tamagawa numbers of motives [3]; it is close in spirit to Scholl’s method of relating non-criticalvalues of L-functions of pure motives to critical values of L-functions of mixed motives [9, p. 252][13, 14]. Notations.
We write A = A f × R for the ring of adeles over Q ; here A f = ˆ Z ⊗ Z Q is the ringof finite adeles. For any number field K , we let O K be the ring of integers, A K denote the ring ofadeles A ⊗ Q K over K ; write I K for the ideles. Let ¯ F be a fixed algebraic closure of F and write = Gal ( ¯
F/F ) for the Galois group of F . For any abelian group P and any integer m > , wewrite P m for the m -torsion subgroup of P . A topological abelian group is Hausdorff. Construction of Y A . This will use the continuous cohomology of Γ via classifying spaces as in[10, 5] to which we refer for a detailed exposition.For each field L with F ⊂ L ⊂ ¯ F , the group G ( A L ) is a locally compact group. If L/F isGalois, then G ( A L ) Gal(
L/F ) = G ( A F ) . So E = lim → G ( A L ) , the direct limit of locally compact abelian groups, is equipped with a continuous action of Γ . Thenatural map(2) E := G ( ¯ F ) ֒ → E is Γ -equivariant. Though the subgroup G ( F ) ⊂ G ( A F ) is discrete, the subgroup E ⊂ E fails to be discrete; this failure happens at finite level (see Lemma 4 below). The non-Hausdorffnature of the quotient E /E directs us to consider the classifying space/topos.Let T op be the site defined by the category of (locally compact) Hausdorff topological spaceswith the open covering Grothendieck topology (as in the "gros topos" of [5, §2]). Any locallycompact abelian group M defines a sheaf yM of abelian groups on T op ; this (Yoneda) provides afully faithful embedding of the (additive, but not abelian) category
T ab of locally compact abeliangroups into the (abelian) category T ab of sheaves of abelian groups on T op . Write T op for thecategory of sheaves of sets on T op and let y : T op → T op be the Yoneda embedding. A generalizedtopology on a given set S is an object F of T op with F ( ∗ ) = S .For any (locally compact) topological group G , its classifying topos B G is the category of objects F of T op together with an action y G × F → F . An abelian group object F of B G is a sheaf on T op , together with actions y G ( U ) × F ( U ) → F ( U ) , functorial in U ; we write H i ( G , F ) (objectsof T ab ) for the continuous/topological group cohomology of G with coefficients in F . These arisefrom the global section functor B G → T ab, F F y G . Details for the following facts can be found in [5, §3] and [10].(a) (Yoneda) Any topological G -module M provides an (abelian group) object yM of B G ; see[10, Proposition 1.1].(b) If → M → N is a map of topological G -modules with M homeomorphic to its image in N , then the induced map yM → yN is injective [5, Lemma 4].(c) Applying Propositions 5.1 and 9.4 of [5] to the profinite group Γ and any continuous Γ -module M provide an isomorphism H i (Γ , yM ) ≃ yH icts (Γ , M ) between this topological group cohomology and the continuous cohomology (computedvia continuous cochains). This is also proved in [10, Corollary 2.4]. or any continuous homomorphism f : M → N of topological abelian groups, the cokernel of yf : yM → yN is well-defined in T ab even if the cokernel of f does not exist in T ab . If f is a mapof topological G -modules, then the cokernel of the induced map yf : yM → yN , a well-definedabelian group object of B G , need not be of the form yP .By (a) and (b) above, the pair of topological Γ -modules E ֒ → E (2) gives rise to a pair yE ֒ → y E of objects of B Γ . Write Y for the quotient object y E yE . As E /E is not Hausdorff (Lemma 4), Y isnot yN for any topological Γ -module N . Definition 1.
We set Y A = H (Γ , Y ) ∈ T ab .Our main result is the Theorem 2. (i) Y A is the Yoneda image yY A of a Hausdorff locally compact topological abeliangroup Y A .(ii) X A is an open subgroup of Y A .(iii) The group Y A is compact if and only if Ш ( A/F ) is finite. If Y A is compact, then the indexof X A in Y A is equal to Ш ( A/F ) . As Ш ( A/F ) is a torsion discrete group, the topology of Y A is determined by that of X A . Proof. (of Theorem 2) The basic point is the proof of (iii). From the exact sequence → yE → y E → Y → of abelian objects in B Γ , we get a long exact sequence (in T ab ) → H (Γ , yE ) → H (Γ , y E ) →→ H (Γ , Y ) → H (Γ , yE ) j −→ H (Γ , y E ) → · · · . We have the following identities of topological groups: H (Γ , yE ) = yG ( F ) and H (Γ , y E ) = yG ( A F ) , and by [5, Lemma 4], yG ( A F ) yG ( F ) ≃ yX A . This exhibits yX A as a sub-object of Y A andprovides the exact sequence → yX A → Y A → Ker( j ) → . If Ker( j ) = y Ш ( A/F ) , then Y A = yY A for a unique topological abelian group Y A because Ш ( A/F ) is a torsion discrete group. Thus, it suffices to identify Ker( j ) as y Ш ( A/F ) . Let E δ denote E endowed with the discrete topology; the identity map on the underlying set provides acontinuous Γ -equivariant map E δ → E . Since E is a discrete Γ -module, the inclusion E → E factorizes via E δ . By item (c) above, Ker ( j ) is isomorphic to the Yoneda image of the kernel ofthe composite map H cts (Γ , E ) j ′ −→ H cts (Γ , E δ ) k −→ H cts (Γ , E ) . Since E and E δ are discrete Γ -modules, the map j ′ identifies with the map of ordinary Galoiscohomology groups H (Γ , E ) j ” −→ H (Γ , E δ ) . The traditional definition [2, Lemma 1.16] of Ш ( G/F ) is as Ker( j ”) . As Ш ( A/F ) ≃ Ш ( G/F )
2, Lemma 1.16], to prove Theorem 2, all that remains is the injectivity of k . This is straightforwardfrom the standard description of H in terms of crossed homomorphisms: if f : Γ → E δ is acrossed homomorphism with kf principal, then there exists α ∈ E with f : Γ → E satisfies f ( γ ) = γ ( α ) − α γ ∈ Γ . This identity clearly holds in both E and E δ . Since the Γ -orbit of any element of E is finite, the lefthand side is a continuous map from Γ to E δ . Thus, f is already a principal crossed (continuous)homomorphism. So k is injective, finishing the proof of Theorem 2. (cid:3) Remark 3.
The proof above shows: If the stabilizer of every element of a topological Γ -module N is open in Γ , then the natural map H (Γ , N δ ) → H (Γ , N ) is injective. Bloch’s semi-abelian variety G . [2, 11]Write A ∨ ( F ) = B × finite. By the Weil-Barsotti formula, Ext F ( A, G m ) ≃ A ∨ ( F ) . Every point P ∈ A ∨ ( F ) determines a semi-abelian variety G P which is an extension of A by G m .Let G be the semiabelian variety determined by B :(3) → T → G → A → , an extension of A by the torus T = Hom ( B, G m ) . The semiabelian variety G is the Cartier dual[4, §10] of the one-motive [ B → A ∨ ] . The sequence (3) provides (via Hilbert Theorem 90) [2, (1.4)] the following exact sequence(4) → T ( A F ) T ( F ) → G ( A F ) G ( F ) → A ( A F ) A ( F ) → . It is worthwhile to contemplate this mysterious sequence: the first term is a Hausdorff, non-compact group and the last is a compact non-Hausdorff group, but the middle term is a compactHausdorff group!
Lemma 4.
Consider a field L with F ⊂ L ⊂ ¯ F . The group G ( L ) is a discrete subgroup of G ( A L ) if and only if A ( K ) ⊂ A ( L ) is of finite index.Proof. Pick a subgroup C ≃ Z s of A ∨ ( L ) such that B × C has finite index in A ∨ ( L ) . The Blochsemiabelian variety G ′ over L determined by B × C is an extension of A by T ′ = Hom ( B × C, G m ) .One has an exact sequence → T ” → G ′ → G → defined over L where T ” = Hom ( C, G m ) isa split torus of dimension s . Consider the commutative diagram with exact rows and columns y y T ”( A L ) T ”( L ) T ”( A L ) T ”( L ) y y −−−→ T ′ ( A L ) T ′ ( L ) −−−→ G ′ ( A L ) G ′ ( L ) −−−→ A ( A L ) A ( L ) y y (cid:13)(cid:13)(cid:13) −−−→ T ( A L ) T ( L ) −−−→ G ( A L ) G ( L ) −−−→ A ( A L ) A ( L ) y y y . The proof of surjectivity in the columns follows Hilbert Theorem 90 applied to T ” [2, (1.4]). TheBloch-Tamagawa space X ′ A = G ′ ( A L ) G ′ ( L ) for A over L is compact and Hausdorff; its quotient by T ”( A L ) T ”( L ) = ( I L L ∗ ) s is G ( A L ) G ( L ) . The quotient is Hausdorff if and only if s = 0 . (cid:3) A more general form of Lemma 4 is implicit in [2]: For any one-motive [ N φ −→ A ∨ ] over F ,write V for its Cartier dual (a semiabelian variety), and put X = V ( A F ) V ( F ) . We assume that the Γ -action on N is trivial. Then X is compact if and only if Ker( φ ) is finite; X is Hausdorff if and only if the image of φ has finite index in A ∨ ( F ) . Tamagawa numbers.
Let H be a semisimple algebraic group over F . Since H ( F ) embeds dis-cretely in H ( A F ) , the adelic space X H = H ( A F ) H ( F ) is Hausdorff. The Tamagawa number τ ( H ) is thevolume of X H relative to a canonical (Tamagawa) measure [15]. The Tamagawa number theorem[8, 1] (which was formerly a conjecture) states(5) τ ( H ) = H ) torsion Ш ( H ) where Pic( H ) is the Picard group and Ш ( H ) the Tate-Shafarevich set of H/F (which measuresthe failure of the Hasse principle). Taking H = SL over Q in (5) recovers Euler’s result ζ (2) = π . The above formulation (5) of the Tamagawa number theorem is due to T. Ono [12, 17] whosestudy of the behavior of τ under an isogeny explains the presence of Pic( H ) , and reduces thesemisimple case to the simply connected case. The original form of the theorem (due to A. Weil)is that τ ( H ) = 1 for split simply connected H . The Tamagawa number theorem (5) is valid, more enerally, for any connected linear algebraic group H over F . The case H = G m becomes theTate-Iwasawa [16, 7] version of the analytic class number formula: the residue at s = 1 of the zetafunction ζ ( F, s ) is the volume of the (compact) unit idele class group J F = Ker( | − | : I F F ∗ → R > ) of F . Here | − | is the absolute value or norm map on I F . Zagier extensions. [18] The m -Selmer group Sel m ( A/F ) (for m > ) fits into an exact sequence(6) → A ( F ) mA ( F ) → Sel m ( A/F ) → Ш ( A/F ) m → . D. Zagier [18, §4] has pointed out that while the m -Selmer sequences (6) (for all m > ) cannotbe induced by a sequence (an extension of Ш ( A/F ) by A ( F ) ) → A ( F ) → ? → Ш ( A/F ) → , they can be induced by an exact sequence of the form(7) → A ( F ) → A → S → Ш ( A/F ) → and gave examples of such (Zagier) sequences. Combining (1) and (4) above provides the follow-ing natural Zagier sequence → A ( F ) → A ( A F ) → Y A T ( A F ) → Ш ( A/F ) → . Write A ( A ¯ F ) for the direct limit of the groups A ( A L ) over all finite subextensions F ⊂ L ⊂ ¯ F .The previous sequence discretized (neglect the topology) becomes → A ( F ) → A ( A F ) → ( A ( A ¯ F ) A ( ¯ F ) ) Γ → Ш ( A/F ) → . Remark 5. (i) For an elliptic curve E over F , Flach has indicated how to extract a canonicalZagier sequence via τ ≥ τ ≤ R Γ( S et , G m ) from any regular arithmetic surface S → Spec O F with E = S × Spec O F Spec F .(ii) It is well known that the class group Pic( O F ) is analogous to Ш ( A/F ) and the unit group O × F is analogous to A ( F ) . Iwasawa [6, p. 354] proved that the compactness of J F is equivalent tothe two basic finiteness results of algebraic number theory: (i) Pic( O F ) is finite; (ii) O × F is finitelygenerated. His result provided a beautiful new proof of these two finiteness theorems. Bloch’sresult [2, Theorem 1.10] on the compactness of X A uses the Mordell-Weil theorem (the group A ( F ) is finitely generated) and the non-degeneracy of the Néron-Tate pairing on A ( F ) × A ∨ ( F ) (modulo torsion). Question 6.
Can one define directly a space attached to
A/F whose compactness implies theMordell-Weil theorem for A and the finiteness of Ш ( A/F ) ? Acknowledgements.
I thank C. Deninger, M. Flach, S. Lichtenbaum, J. Milne, J. Parson, J. Rosen-berg, L. Washington, and B. Wieland for interest and encouragement and the referee for helpingcorrect some inaccuracies in an earlier version of this note. I have been inspired by the ideas ofBloch, T. Ono (via B. Wieland [17]) on Tamagawa numbers, and D. Zagier on the Tate-Shafarevichgroup [18]. I am indebted to Ran Cui for alerting me to Wieland’s ideas [17]. EFERENCES [1] Aravind Asok, Brent Doran, and Frances Kirwan. Yang-Mills theory and Tamagawa numbers: the fascination ofunexpected links in mathematics.
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